Place the frozen strawberries, ⅓ cup sugar, and cornstarch in a small saucepan. Set the heat to medium-low, stirring occasionally, until the berries have softened. This should take between 5 and 10 minutes.
When the berries have mostly thawed, increase the heat to medium. Mash them a bit with a potato masher and stir constantly while the liquid simmers and the sauce thickens, about 2 minutes more. Remove from the heat and cool completely before placing in the refrigerator to chill until needed.
Ice Cream Instructions
Whisk together the cream, milk, sugar, salt, and vanilla extract. Pour into the ice cream machine and freeze according to the manufacturer's directions.
Scoop the churned ice cream into a freezer container, adding a spoonful or two of the strawberry sauce along with each scoop of the ice cream. Scatter the cookie pieces throughout the ice cream while transferring it to the container.
Swirl very lightly with a knife and place in the freezer until ready to serve. Serve with any remaining strawberry sauce on top. Add a dollop of whipped cream as well, if desired.
Notes
Fresh strawberries may be used to make the sauce for this recipe. If using fresh berries, reduce the cooking time accordingly. As soon as they begin to break down and the sauce thickens, remove it from the heat.